Wednesday, March 26, 2014

Compound Inequalities Treasure Map

Never underestimate the power of novelty to help you engage certain students.

I just spent the last hour and a half-long block period with my jaw on the floor, watching in amazement as my most discouraged, 12th grade College Prep Math students worked productively and peacefully on, of all things, the analysis and solving of compound inequalities.

During my prep, I turned a boring worksheet into a treasure map. And that turned a boring requirement into a very peaceful and enjoyable period.

As she was leaving, one girl asked, Could we please do more work like this?

I'll take that as a compliment!

Sunday, March 16, 2014

Stalkers and dreamers

I've talked about this before: there are those who learn by stalking — step by step, one day at a time, one skill at a time, little by little. And then there are dreamers: those of us who try and fail, try and fail, try and fail. Carlos CastaƱeda makes this distinction between stalkers and dreamers, and it has been a useful distinction for me from the moment I encountered it.

I am a dreamer. I first became aware of this learning pattern when I was about five and learning to ride on two wheels. My dad removed the training wheels from my little red bike and I would practice.

I practiced riding day after day for weeks.

And day after day, for weeks on end, I would fail.

I fell everywhere — on the sidewalk in front of our house, in the driveway, on our block at low-traffic times.

I fell on smooth pavement, on concrete, any time I encountered gravel.

I was frustrated and pretty scuffed up. But in my mind's eye at night, I could imagine riding perfectly.

When I dreamed, I could feel myself rolling smoothly and swiftly on two wheels. In my dream life, I was a person who could ride on two wheels, and I could do so successfully anywhere.

A little less so in my waking life.

I skidded, slid, or toppled over after only a few feet. I still remember how it felt to fall at different moments and on different surfaces. I have a vivid and complete felt sense memory of rolling onto a patch of gravel and sliding to the ground at the intersection of Lenox Road and Hershey. I distinctly remember the feeling of gravel biting into the skin of my knee.

I must have wanted it bad to keep on trying.

Then one day, it just happened.

I had steeled myself yet again for the failure that had become my 'normal,' and I readied myself for more cuts and bruises and wounded ego.

But I didn't fall over.

I was so excited I parked my bike and the garage and rushed inside to tell my mother what had happened. It was the most exhilarating thing I could imagine at the time.

Still, though, I assumed it was a fluke. I continued to prepare myself for further failure.

But then it happened again. And again. And again.

The story of the larva that becomes a butterfly had taken hold of my life. Even after you emerge from the transformation, it takes a while for your awareness to catch up with your changed reality. It took several days before I realized I had stepped into a new normal.

I'd been afraid to hope.

Nowadays I wonder how my students experience transformation from people who believe they can't do math into people who understand that they can. It's hard to trust transformation. As A.H. Almaas says, you've been a larva crawling around all your life, and you believe that the best you can hope for is to crawl faster and to become a bigger, fatter, happier, more successful larva. You see butterflies flying around and you classfy them as anomalies. Most of us never automatically think, gee, that's where *I'm* headed too. Most people think, "Wow those are really interesting beings. I wonder where they come from. I wonder what it would feel like to be one of those."

In math, as in learning to ride a bicycle, it never occurred to me that I could take what I know from other areas of my life to help myself become one of those magical creatures who can ride a bicycle or do math. I did not know it was my birthright to be good and successful at those things. I thought I was destined to remain an earthbound larva.

For a lot of us, it's not enough to say, if you can't do these problems fluently after this investigation, then that means you need to seek out more practice. I needed both experience or discovery and also practice. I needed opportunities for practice and maybe a choice of activities that allowed me to seek out the practice I needed while others were ready for more discovery. Maybe this takes the form of a branching of activities — a practice table and an extensions table, for example. All I know is that students need support and opportunities to self-diagnose and to seek out the experiences they need in that moment. Stalkers need space to stalk further while dreamers need space and time to practice and fall down a lot more.

There is a mystical part of this process that cannot be discounted. 

At times when I feel discouraged about my teaching practice, I have to remind myself about all of this. I feel like I am trying and failing, trying and failing, trying and failing. I have a lot of psychic gravel chewing through the skin on my psychic knees from falling down. I have to remind myself that this is my process.

Sunday, March 2, 2014

Attending to Precision: INBs and group work (Interactive Notebooks)

I love new beginnings, but I am only so-so with early middles. Now that kids have started their INB journey, we've arrived at that crucial moment between the beginning and the first INB check. This, as the saying goes, is where the rubber meets the road.

I find that kids never understand at this stage why I insist on being so darned nit-picky about their notebooks. Every day someone new asks me why this or that HAS to go on the right-hand side or EXACTLY on page 5.

One of the many reasons why this is important, I have learned, is that it is all about teaching strategies for attending to precision — Mathematical Practice Standard #6, which is defined this way in the standards documents:
Mathematically proficient students try to communicate precisely to others.• They try to use clear definitions in discussion with others and in their own reasoning.When making mathematical arguments about a solution, strategy, or conjecture (see MP.3), mathematically proficient elementary students learn to craft careful explanations that communicate their reasoning by referring specifically to each important mathematical element, describing the relationships among them, and connecting their words clearly to their representations. They state the meaning of the symbols they choose, including us- ing the equal sign consistently and appropriately.• They are careful about specifying units of measure,• and labeling axes to clarify the correspondence with quantities in a problem.• They calculate ac- curately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 
The problem, I find, is that this description of precision is precise only at the theoretical level. On the front lines, it's unrealistic because most kids never get to this level of precision.

And that is because their notes and their work are generally quite a mess.

A big part of teaching students to attend to precision is giving them a structure for being an impeccable warrior as a math student — that is to say, taking and keeping good notes, noticing and keeping track of your own progress as a learner, preserving your homework in a predictable place that is not, let us say, the very bottom of your backpack, crushed into a handful of loose raisins.

It means stepping up your game as a student of mathematics and presenting your work in a way that makes it possible for others to notice the care with which you are specifying units, crafting careful explanations, describing relationships, and so on. And it means presenting your work in this way ALWAYS — in all things, in all times, wherever you go.

INBs are an incredibly low-barrier-to-entry, accessible structure for teaching attention to precision. There are no students who cannot benefit from having a clear, common, and predictable structure for organizing their learning. INBs are also a great leveler. For those of us who are focused on creating equity in our classrooms, INBs offer all students a chance to prove both to themselves and others that they are indeed smart in mathematics. As I saw the other night at Back To School Night, my strongest note-keeping students are rarely the top students computationally speaking. But they are the ones who can always find what they are looking for — a major advantage on an open-notes test.

INBs are also a phenomenal formative assessment tool. Flipping through a students INB gives me an incredible snapshot of where and when they were truly attending to precision and where they were fuzzing out. Blank spaces and lack of color or highlighter on specific notes pages give me a targeted spot for further formative assessment. In my experience, it is exceedingly rare for a student who thoroughly understands a topic to write no notes or diagrams on that page. If anything, they are the ones who are most likely to appreciate the chance to consolidate their understanding.

So I am sticking with it and zooming in on some of the areas where kids' understanding fell apart last week. We'll be reviewing how to convert from percentages to decimals and how to document and analyze the iterative process of calculating compound interest because that is where my students' notes fell apart.

I'll be astonished — but will report back honestly — if these on-the-fly assessments prove to have been inaccurate.

Monday, February 24, 2014

New strategy for introducing INBs: complex instruction approach

After months of not feeling like my best teacher self in the classroom, I got fed up and spent all weekend tearing stuff down and rebuilding from the ground up.

INBs are something I know well — something that work for students. So I decided to take what I had available and, as Sam would say, turn what I DON'T know into what I DO know. Love those Calculus mottos.

So I rebuilt my version of the exponential functions unit in terms of INBs. But that meant, I would have to introduce INBs.

As one girl said, "New marking period, new me!" The kids just went with it and really took to it.

Here is what I did.

ON EACH GROUP TABLE: I placed a sample INB that began with a single-sheet Table of Contents (p. 1), an Exponential Functions pocket page (p. 3), and had pages numbered through page 7. There were TOC sheets and glue sticks on the table.

SMART BOARD: on the projector, I put a countdown timer (set for 15 minutes) and an agenda slide that said,

  • New seats!
  • Choose a notebook! Good colors still available!
  • Make your notebook look like the sample notebook on your table 

As soon as the bell rang, I hit Start on the timer, which counted down like a bomb in a James Bond movie.

Alfred Hitchcock once said, if you want to create suspense, place a ticking time bomb under a card table at which four people are playing bridge. This seemed like good advice for introducing INBs to my students.

I think because it was a familiar, group work task approach to an unfamiliar problem, all the kids simply went went with it. "How did you make the pocket? Do you fold it this way? Where does the table of contents go? What does 'TOC' mean? What goes on page 5?" And so on and so on.

I circulated, taking attendance and making notes about participation. When students would ask me a question about how to do something, I would ask them first, "Is this a group question?" If not, they knew what was going to happen. If it was, I was happy to help them get unstuck.

Then came the acid test: the actual note-taking.

I was concerned, but they were riveted. They felt a lot more ownership over their own learning process.

There are still plenty of groupworthy tasks coming up, but at least now they have a container for their notes and reflection process.

I'm going to do a "Five Things" reflection (trace your hand on a RHS page and write down five important things from the day's lesson or group work) and notes for a "Four Summary Statements" poster, but I finally feel like I have a framework to help kids organize their learning.

I've even created a web site with links to photos of my master INB in case they miss class and need to copy the notes. Here's a link to the photo files, along with a picture of page 5:

We only got through half as much as I wanted us to get through, but they were amazed at how many notes we had in such a small and convenient space.

It feels good to be back!

Monday, February 10, 2014

Sometimes the most important thing we say is...

I love this journey I am on in my new school. I love my colleagues and our local community and my administration and the security staff and the office support staff. I love our community outreach coordinators and our special ed department and our paraprofessionals.

But these are not the most important things I get to say on any given day.

The most important thing I ever get to say is something I actually got to say twice today — once at the beginning of first period in a student-parent-staff conference and at the end of the day as I was about to pull out of the parking lot.

I said it to two different students who are each on their own completely different journey right now.

Each had been missing from class for their own different reasons, but they both needed to hear this from me for the same reason.

What I told them was this:
when you are not there, you are missed. 
We miss your voice and your insights and your completely unique presence in the room.

We miss you because we need you.

Whenever you come back to us, we are happy to see you again because you bring something to the community that we need — something we cannot get from any other source.

We need you.

It never ceases to amaze me how many obstacles and complexities and self-defeating behaviors this kind of direct statement clears away.

When you find the opening to say something like this to an adolescent, take it. It's like dropping a pebble into a still pond.

You'll be amazed how far those ripples go.

Saturday, February 8, 2014

Arithmetic of Complex Numbers Placemat Activity - Algebra 2 + Complex Instruction (CI)

Just because you have an all-groupwork and all-Complex Instruction (CI) format doesn't mean you don't need practice activities too.

Our Algebra 2 kids were getting the concepts of complex numbers and complex conjugates, but were still kinda shaky in terms of fluency in working with them.

Based on ideas I stole borrowed a long time ago from the fabulous Kate Nowak (@k8nowak, and the equally fabulous Rachel Kernodle (@rdkpickle, ), I proposed a placemat activity to my ever-game Algebra 2 teaching team and they dove right in.

We have typical CI four-person table teams set up in each of our rooms, with each person assigned a specific role based on where they're seated at the table. Our roles are Facilitator, Resource Manager, Recorder/Reporter, and Team Captain, although of course, your mileage may vary. Each role has specific tasks they are expected to perform; for example, only the Resource Manager may call the teacher over for a group check-in or a group question (in our program, teachers only accept and answer group questions).

Each table was given:
  • two, double-sided "placemat" sheets for doing work in the center of the table
  • a set of problem cards (there are four sets, one for each round of play; to simplify clean-up and organization, I printed each round of cards single-sided on a different color of paper, one set per table group. I've got 7 tables in my room, so I made seven sets of cards. I laminated them and clipped them together, but hey, that's just me)
  • the sum to which all four answers for any given round should add up
The sum for each round was written on the whiteboard, though it could have been projected via document camera or Keynote/Powerpoint slide.

We had mathematical objectives for the activity as well as CI or norms-based, group work objectives. My students in particular needed reinforcement in group work norms and collaboration. Our objectives were:

     Math Objectives

  • achieve greater fluency in the arithmetic of complex numbers (including the distributive property)
  • deepen understanding of and fluency with the powers of i
  • deepen understanding of and fluency with complex conjugates

     Group Work Objectives

  • work in the middle of the table
  •  same problem, same time (no one moves on until everyone moves on)
  • using table group members as resources

The next time I run this activity, I will definitely give a Participation Quiz because the group work norms are so beautifully reinforced in this activity.

How We Ran It
Recorder/Reporter writes the sum in the central oval of the first side of the placemat. Each group member gets a problem card for round 1 (problem a) and works his or her problem on his or her quadrant of the placemat.

When everybody is finished with their problem, the Facilitator facilitates the addition of all four answers. If they add up to the given sum for that round, the Resource Manager calls the teacher over for a "checkpoint" and the next set of cards for the subsequent round of work.

If their answers don't add up to the given sum, they need to work together through everybody's work on the placemat to diagnose what went wrong and where, as well as how to fix it. Then when they've fixed it, they call the teacher over for a checkpoint and the next set of cards for the subsequent round.

Group Work Benefits — Reinforcing Norms
For my classes, the greatest benefits of this activity came from the fact that it forced students to work in the middle of the table, to use each other as resources, and to talk mathematics. Getting kids to work in the middle of the table is the hardest part of CI, in my view, because it goes against the grain of most of their in-school conditioning. The placemat format makes it nearly impossible NOT to work in the middle of the table. And once they're doing that, it seemed like everything else ran pretty smoothly.

I especially liked the fact that this activity created a context in which students experienced an intellectual need for the using the rules of arithmetic for complex numbers and for the powers of i. It was situationally motivated, but extremely targeted.

Sums for Each Round 
The sums for each round are as follows (if you find an error, please speak up):
  • Round 1 (problem a):   26-73i
  • Round 2 (problem b):  0
  • Round 3 (problem c):  165
  • Round 4 (problem d):  2 – 48i
PDF Files for the activity
These are available also on the Math Teacher Wiki on the Algebra 2 page. If you haven't visited the Math Teacher Wiki, you don't know what you're missing.

Tuesday, February 4, 2014

Building concept maps is harder than it looks

I'm having students create a concept map as a summative assessment for our Complex Numbers unit and... w o w — there is all kinds of learning going on.


Students are working in groups and can use all their notes and assignments from the unit. Some kids jumped right in and started hacking away. Others whined and asked why we couldn't just have a normal test.

We are using Post-Its, scissors, pencils, and paper to do our constructions.

In-process projects range from amazing to struggling, but what impresses me most is how much the work reveals about what students are figuring out and how each student is understanding and constructing meaning in their learning. It also demands that learners own their own learning.

Because this is so revealing, I am probably going to use concept maps both as formative assessments before and during the unit as well as using them as a 'ways of understanding' tool to help them consolidate their learning.

BREAKING: OK, this activity is definitely a keeper. Students are really digesting their learning, talking about it, debating how to represent it, and clarifying areas of confusion for themselves. Here is an outstanding example from today: